Prime knot

From Academic Kids

In knot theory, a prime knot is a knot which is, in a certain sense, indecomposable. Specifically, it is one which cannot be written as the knot sum of two nontrivial knots. Knots which are not prime are said to be composite. It can be a nontrivial problem to determine whether a given knot is prime or not.

The nicest examples of prime knots are called torus knots. These are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers.

The simplest prime knot is the trefoil with 3 crossings. The trefoil is actually a (2,3)-torus knot. The figure-eight knot, with 4 crossings, is the simplest non-torus knot. For any positive integer n, there are a finite number of prime knots with n crossings. The first few values are given in the following table.

n 1 2 3 4 5 6 7 8 9 10
Number of prime knots
with n crossings
0 0 1 1 2 3 7 21 49 165

Note that enantiomorphs are counted only once in this table (i.e. a knot and its mirror image are considered equivalent).

Small prime knots
3 Missing image
Knot-trefoil-left-64.png
Trefoil

4 Missing image
PrimeKnot-4-1.png
Figure-8 knot

5 Cinquefoil Missing image
PrimeKnot-5-2.png
Image:PrimeKnot-5-2.png

6 Missing image
Knot-stevedore-sm.png
Stevedore's knot

Missing image
PrimeKnot-6-2.png
Image:PrimeKnot-6-2.png

Missing image
PrimeKnot-6-3.png
Image:PrimeKnot-6-3.png

External links

  • Prime Knot (http://mathworld.wolfram.com/PrimeKnot.html) at MathWorld.
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